Conclusion — Definition, Meaning & Examples
Conclusion
The part of a conditional statement after then.
For example, the conclusion of "If a line is horizontal then the line has slope 0" is "the line has slope 0".
See also
Hypothesis, converse, inverse, contrapositive, inverse of a conditional, slope
Example
Problem: Identify the hypothesis and conclusion in the conditional statement: "If a triangle has three equal sides, then it is equilateral."
Step 1: Locate the "if" and "then" in the statement. The structure is: If [hypothesis], then [conclusion].
Step 2: The hypothesis is the clause after "if": "a triangle has three equal sides."
Step 3: The conclusion is the clause after "then": "it is equilateral."
Step 4: In symbolic form, if we let p = "a triangle has three equal sides" and q = "it is equilateral," the conditional is written as:
p→q
Answer: The conclusion is "it is equilateral." This is the part of the conditional that is claimed to be true whenever the hypothesis holds.
Another Example
Problem: Rewrite the statement "All right angles measure 90°" as a conditional, then identify the conclusion.
Step 1: Not every conditional statement is written in obvious "if-then" form. Rewrite the sentence: "If an angle is a right angle, then it measures 90°."
Step 2: The hypothesis (after "if") is: "an angle is a right angle."
Step 3: The conclusion (after "then") is: "it measures 90°."
Answer: The conclusion is "it measures 90°." Notice that the original sentence had no explicit "if" or "then," but once rewritten as a conditional, the conclusion becomes clear.
Frequently Asked Questions
What is the difference between a hypothesis and a conclusion in math?
The hypothesis is the "if" part of a conditional statement — it states the condition or assumption. The conclusion is the "then" part — it states what follows when the hypothesis is satisfied. For example, in "If x > 5, then x > 3," the hypothesis is "x > 5" and the conclusion is "x > 3."
Can a conclusion be false even if it comes after 'then'?
Yes. A conditional statement claims the conclusion is true whenever the hypothesis is true. If the hypothesis is false, the entire conditional is still considered true in logic regardless of whether the conclusion is true or false. The conclusion itself is only guaranteed to hold when the hypothesis is met.
Hypothesis vs. Conclusion
| Hypothesis | Conclusion | |
|---|---|---|
| Role in conditional | The 'if' part (the condition assumed true) | The 'then' part (the result that follows) |
| Example: If a polygon is a square, then it has four right angles | A polygon is a square | It has four right angles |
| Swapping creates | Converse (may or may not be true) | Inverse (may or may not be true) |
| Dimension | Sets up the scenario | States what follows from it |
Why It Matters
Identifying the conclusion correctly is essential for writing proofs, especially in geometry, where you must show that the conclusion follows logically from given information. It also matters when you form related statements like the converse, inverse, and contrapositive — each rearranges or negates the hypothesis and conclusion in a specific way. Misidentifying which part is the conclusion leads to errors in all of these logical operations.
Common Mistakes
Mistake: Confusing the hypothesis and conclusion when the statement is not written in standard "if-then" form.
Correction: Always rewrite the statement in "If ___, then ___" form first. The part after "then" is the conclusion. For instance, "Parallel lines never intersect" becomes "If two lines are parallel, then they never intersect" — the conclusion is "they never intersect."
Mistake: Assuming the conclusion is always true on its own.
Correction: The conclusion is only guaranteed to be true when the hypothesis is satisfied. The statement "If a number is even, then it is divisible by 2" does not mean every number is divisible by 2 — only even numbers are.
Related Terms
- Conditional — The if-then statement containing a conclusion
- Hypothesis — The "if" part paired with the conclusion
- Converse — Swaps the hypothesis and conclusion
- Inverse of a Conditional — Negates both hypothesis and conclusion
- Contrapositive — Negates and swaps hypothesis and conclusion
- Line — Common subject in geometric conditionals
- Horizontal — Used in the classic slope-zero example